Problem: Given $ m \angle QPR = 2x + 5$, $ m \angle RPS = 4x + 13$, and $ m \angle QPS = 120$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {2x + 5} + {4x + 13} = {120}$ Combine like terms: $ 6x + 18 = 120$ Subtract $18$ from both sides: $ 6x = 102$ Divide both sides by $6$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 2({17}) + 5$ Simplify: $ {m\angle QPR = 34 + 5}$ So ${m\angle QPR = 39}$.